Complex solutions to the wave equation

Explain why, for a sinusoidal pulse incoming from the left and moving to the right, the solution to the wave equation has the form

$$\begin{array}{l} y_0(x\le0,t)=\mbox{Re\,}\left[ \underline{A}... ...ega t} \left( \underline{T} e^{ikx} \right)\right]. \end{array}$$ (1)

Your explanation can be simply the identification of the meaning of each term in each equation.